2.5 Transformations of Functions

2.5 Transformations of
Functions
Chapter 2 Functions and Graphs

Concepts and Objectives
⚫ Objectives for this section are
⚫ Recognize graphs of common functions.
⚫ Graph functions using vertical and horizontal shifts.
⚫ Graph functions using reflections about the x-axis and
the y-axis.
⚫ Graph functions using compressions and stretches.
⚫ Combine transformations.

“Toolkit Functions”
⚫ When working with functions, it is helpful to have a base
set of building-block elements.
⚫ We call these our “toolkit functions,” which form a set of
basic named functions for which we know the graph,
formula, and special properties.
⚫ For these definitions we will use x as the input variable
and y = f(x) as the output variable.
⚫ We will see these functions and their combinations and
transformations throughout this course, so it will be
very helpful if you can recognize them quickly.

Constant Function f(x) = c
⚫ f(x) = c is constant and continuous on its entire domain.
⚫ Even function
Domain: (–∞ ∞) Range: [c, c]
x y
–2 c
–1 c
0 c
1 c
2 c
c

Identity Function f(x) = x
⚫ f(x) = x is increasing and continuous on its entire domain.
⚫ Odd function
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
–2 –2
–1 –1
0 0
1 1
2 2

Absolute Value Function
⚫ decreases on the interval (–∞ 0] and
increases on the interval [0, ∞).
⚫ Even function
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
-2 2
-1 1
0 0
1 1
2 2
( )
f x x
=
( )
f x x
=

Quadratic Function f(x) = x2
⚫ f(x) = x2 decreases on the interval (–∞ 0] and increases
on the interval [0, ∞).
⚫ Even function
Domain: (–∞ ∞) Range: [0 ∞)
x y
–2 4
–1 1
0 0
1 1
2 4
vertex

Square Root Function
⚫ increases and is continuous on its entire
domain.
⚫ Neither even nor odd
Domain: [0 ∞) Range: [0 ∞)
x y
0 0
1 1
4 2
9 3
16 4
( )
f x x
=
( )
f x x
=

Cubic Function f(x) = x3
⚫ f(x) = x3 increases and is continuous on its entire domain.
⚫ Odd function
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
–2 –8
–1 –1
0 0
1 1
2 8

Cube Root Function
⚫ increases and is continuous on its entire
domain.
⚫ Odd function
Domain: (–∞ ∞) Range: (–∞ ∞)
x y
-8 -2
-1 -1
0 0
1 1
8 2
( ) 3
f x x
=
( ) 3
f x x
=

Translations
⚫ Notice the difference between the graphs of
and and their parent graphs.
( )
y f x c
= +
( )
y f x c
= +
( )
f x x
= ( )
f x x
=
( ) 3
g x x
= − ( ) 3
h x x
= −

Translations (cont.)
• If a function g is defined by g(x) = f(x) + k, where k is
a real number, then the graph of g will be the same
as the graph of f, but translated |k| units up if k is
positive or down if k is negative.
Vertical Translations (Shift)
• If a function g is defined by g(x) = f(x – h), where h is a
real number, then the graph of g will be the same as
the graph of f, but translated |h| units to the right if
h is positive or left if h is negative.
Horizontal Translations (Shift)

⚫ Why does y shift the function to the right?
Shouldn’t a negative number go to the left?
⚫ Consider the function . It represents a
horizontal translation of the square function. Look at
the table and graph:
( )
2
y f x
= −
( )
2
2
y x
= −
x x – 2 y
2 0 0
3 1 1
4 2 4

⚫ Let’s look at the graph more closely:
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
2 2
2 2
2 2
4 2 4 2 4
1 1 3 2 3
0 0 2 2 2
y f
y f
y f
= = = − =
= = = − =
= = = − =

Reflecting a Graph
⚫ Forming a mirror image of a graph across a line is called
reflecting the graph across the line.
⚫ Compare the reflected graphs and their parent graphs:
y x
=
y x
= −
y x
=
y x
= −
 over x-axis
over y-axis →

Reflecting a Graph (cont.)
⚫ The graph of y = –f(x) is the same as the graph of y = f(x)
reflected across the x-axis.
⚫ The graph of y = f(–x) is the same as the graph of y = f(x)
reflected across the y-axis.

Stretches and Compression
⚫ Compare the graphs of g(x) and h(x) with their parent
graphs:
( ) 2
g x x
=
• Narrower
• Same x-intercept
• Vertical stretching

Stretches and Compression (cont.)
( )
1
2
h x x
=
• Wider
• Vertical
compression

⚫ These graphs show the distinction between
and :
( )
y af x
=
( )
y f ax
=
( )
y f x
=
( )
2
y f x
=
( )
y f x
=
( )
2
y f x
=
• Different y-intercept
• Different x-intercept
• Same y-intercept
vertical stretching horizontal compression

Generally speaking, if a ≠ 0, then

⚫ The simplest way I’ve found to keep these
transformations straight is to compare them to
translations:
⚫ If a is outside the function, then it is a vertical stretch
or compression.
⚫ If a is inside the function, then it is a horizontal
stretch or compression.
⚫ Also,
⚫ If |a| is greater than 1, it is a stretch; if |a| is less than
1, it is a compression.

Summary
Function Graph Description
y = f(x) Parent function
y = af(x) Vertical stretch (|a| > 1) or compression (0 < |a| < 1)
y = f(ax) Horizontal stretch (0 < |a| < 1) or compression (|a| > 1)
y = –f(x) Reflection about the x-axis
y = f(–x) Reflection about the y-axis
y = f(x) + k Vertical translation up (k > 0) or down (k < 0)
y = f(x – h) Horizontal translation right (h > 0) or left (h < 0)
Note that for the horizontal translation, f(x – 4) would translate the
function 4 units to the right because h is positive.

Combinations
⚫ Example: Given a function whose graph is y = f(x),
describe how the graph of y = –f(x + 2) – 5 is different
from the parent graph.

Combinations
⚫ Example: Given a function whose graph is y = f(x),
describe how the graph of y = –f(x + 2) – 5 is different
from the parent graph.
The graph is translated 2 units to the left, 5 units down,
and the entire graph is reflected across the x-axis.

For Next Class
⚫ Section 2.5 in MyMathLab
⚫ Quiz 2.5 in Canvas
⚫ Optional: Read section 2.6

2.5 Transformations of Functions

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2.5 Transformations of Functions